3.44. Add Your Own Distribution to @RISK

Applies to: @RISK 5.0 and newer

I need a particular distribution that isn't in the Define Distributions dialog. Can I just give @RISK a formula for the CDF and have @RISK draw the random numbers?

If you have a formula for the inverse CDF, you can use it with @RISK to create your own distribution. The input to that formula is a RiskUniform(0,1), which provides a randomly selected cumulative probability; then your inverse CDF formula converts that to an x value. By enclosing the formula in RiskMakeInput( ), you tell @RISK to treat the formula as a regular distribution for purposes like sensitivity analysis and graphing.

We'll illustrate this with the Burr distribution. (Starting with release 7.5, the Burr12 distribution is built into @RISK, but you would use the same method if you need to create a distribution that's not in @RISK.) Wikipedia gives the CDF of a Burr Type XII as

F(x; c,k) = 1 - (1 + xc)-k

where c and k are positive real numbers. A little algebra gives the inverse as

x = [ (1-F)-1/k - 1 ]1/c

To draw random numbers for its standard distributions, @RISK first draws a random number from ithe uniform distribution 0 to 1, which represents a cumulative probability; then it finds the x value corresponding to that percentile — in other words, it uses that cumulative probability as input to an inverse CDF. (@RISK uses special techniques for distributions that don't have a closed form for their inverse CDFs.) Therefore, your Excel formula for a Burr distribution is the combination of RiskUniform and the inverse CDF above:

=( (1-RiskUniform(0,1))^(-1/k) - 1 )^(1/c)

Finally, you want to wrap that in a RiskMakeInput, so that @RISK will store iteration values of this formula, let you make graphs, treat it as an input in sensitivity analyses, and so on. Your final Excel formula is:

You'll replace the parameters c and k with numbers, or more likely with cell references.

To see the formula in action, open the attached workbook in @RISK. The four graphs were made with the four combinations of c and k shown in the worksheet; you can compare these to the PDF curves show in the Wikipedia article. You can also enter your desired values of c and k in any of columns A through D, and run a simulation.